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Equidistribution of cusp points of Hecke triangle groups

Laura Breitkopf, Marc Kesseböhmer, Anke Pohl

Abstract

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural generators. This is a generalization of the corresponding results for the modular group, for which we rely on advanced results from infinite ergodic theory and transfer operator techniques developed for AFN-maps.

Equidistribution of cusp points of Hecke triangle groups

Abstract

In the framework of infinite ergodic theory, we derive equidistribution results for suitable weighted sequences of cusp points of Hecke triangle groups encoded by group elements of constant word length with respect to a set of natural generators. This is a generalization of the corresponding results for the modular group, for which we rely on advanced results from infinite ergodic theory and transfer operator techniques developed for AFN-maps.
Paper Structure (12 sections, 19 theorems, 163 equations, 3 figures)

This paper contains 12 sections, 19 theorems, 163 equations, 3 figures.

Table of Contents

  1. Introduction and statement of main results
  2. Main results
  3. Key elements of proofs, and structure of article
  4. Related results
  5. Fundamental properties of the generalized Farey maps
  6. Origin of the generalized Farey maps
  7. Topologically mixing
  8. AFN-maps
  9. Transfer operators and invariant measure
  10. Tail probabilities
  11. Proof of Theorem \ref{['thm:dyn_intervals']}
  12. Proof of Theorems \ref{['thm:jac_weight_lim_intro']} and \ref{['thm:weighted_distr']}

Key Result

Theorem 1.1

Let $q\geq 5$ odd. For all $0<\alpha\leq\beta\leq 1$ we have

Figures (3)

  • Figure 1: The Stern--Brocot tree restricted to $[0,1]$, starting at tree level $1$.
  • Figure 2: Graph of the generalized Farey map associated to $q=5$
  • Figure 3: First elements of generalized Stern--Brocot sequence $(\mathcal{S}_{n,5})$, omitting $\mathcal{S}_{0,5} = \{0,1\}$

Theorems & Definitions (37)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Special case of melbourne_operator_2012
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 27 more